When you encode the search tree as a proof, is there a technical name for that technique?
In my situation I encoded the proof as a sequence which is just the depth-first linear labelling of the tree. Then I proved why the sequence represents a proof. But if there is a most standard terminology/literature on this step, it would make it easier for me to talk about.
One thing worth trying to do (if you can) is separate the generation of the tree from the tree itself -- hopefully you can imagine the tree existing as a complete object (even though you never store it all), and formulate proofs based on that.
There is also the "proof by certificate" approach. You generate a potentially large proof certificate by an unverified program. Then you run a proven correct (small) checker on the certificate to concude that the theorem is correct.
With the right spin anything can get into the media. It helps if no-one used that before; your 63000 are far more impressive but won't get much traction after this. However it is summer now so the coming months it'll be quite easy to get mostly anything in the news(paper).
This problem has much more than just a search tree. They had to check 10 to 2300 possibilities to proof their counter example, so a simple search tree was no option and they had to boil it down to "just under 1 trillion". So lots of maths involved here.
There is actually 12,418,001,077,381,302,684 semigroups of size 10. We only had to find 718,981,858,383,872 by brute force search (which still involved a whole bunch of maths and case splitting), which is 0.006% of the total. There is a whole PhD in the maths involved.
Of course, we could (and have) just published the search which has to be redone, and the maths which is required to prove the piece of search to be done.
(Personally I love semigroups, and I think it's cool that you counted all the ones of size 10. But any story about them in the popular press would focus on the sheer size of the calculation rather than what it actually achieved.)
However, for numbers of the size I was looking at, storing them is impossible. We were just interested in how many there were (which wasn't known, and we don't know how to calculate without enumerating them through search).
I had a much larger "proof", where we didn't bother storing all the details, in which we enumerated 718,981,858,383,872 semigroups, towards counting the semigroups of size 10.
Uncompressed, it would have been about 63,000 terabytes just for the semigroups, and about a thousand times that to store the "proof", which is just the tree of the search.
Of course, it would have compressed extremely well, but also I'm not sure it would have had any value, you could rebuild the search tree much faster than you could read it from a disc, and if anyone re-verified our calculation I would prefer they did it by a slightly different search, which would give us much better guarantees of correctness.